| L(s) = 1 | + 2·7-s + 4·17-s − 12·23-s + 6·25-s + 8·31-s + 20·41-s + 16·47-s + 3·49-s + 4·71-s − 12·73-s + 16·79-s − 4·89-s + 4·97-s + 16·103-s − 32·113-s + 8·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 24·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.970·17-s − 2.50·23-s + 6/5·25-s + 1.43·31-s + 3.12·41-s + 2.33·47-s + 3/7·49-s + 0.474·71-s − 1.40·73-s + 1.80·79-s − 0.423·89-s + 0.406·97-s + 1.57·103-s − 3.01·113-s + 0.733·119-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 1.89·161-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.834594431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.834594431\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.471838962770586415948708454519, −8.200686335427796865503521379987, −7.80381142093737616370992922728, −7.78688872305237803604980339164, −7.08969589350899637640021816754, −7.01686402147851736568768611083, −6.23009748953773225726142378952, −5.96576646339378720692679316871, −5.80251375175720343170288047653, −5.35845568595979807839408016799, −4.73007558741843598847849267318, −4.50793801986987985479109833042, −4.02328660278433705918852372644, −3.84443787968040537033968804552, −3.06740082358320647091756732469, −2.69463069805405947864151000994, −2.22795057727428517874093116653, −1.78423328056731611659488669630, −0.950956034959552966560564407694, −0.70999647413327453779454816643,
0.70999647413327453779454816643, 0.950956034959552966560564407694, 1.78423328056731611659488669630, 2.22795057727428517874093116653, 2.69463069805405947864151000994, 3.06740082358320647091756732469, 3.84443787968040537033968804552, 4.02328660278433705918852372644, 4.50793801986987985479109833042, 4.73007558741843598847849267318, 5.35845568595979807839408016799, 5.80251375175720343170288047653, 5.96576646339378720692679316871, 6.23009748953773225726142378952, 7.01686402147851736568768611083, 7.08969589350899637640021816754, 7.78688872305237803604980339164, 7.80381142093737616370992922728, 8.200686335427796865503521379987, 8.471838962770586415948708454519
