| L(s) = 1 | + 1.61·3-s − 1.17·5-s + 1.90·7-s + 1.61·9-s + 0.618·11-s − 1.90·15-s + 3.07·21-s − 1.17·23-s + 0.381·25-s + 27-s − 1.90·29-s + 1.00·33-s − 2.23·35-s + 1.90·37-s − 1.61·43-s − 1.90·45-s + 2.61·49-s − 0.726·55-s + 3.07·63-s − 1.90·69-s − 1.61·73-s + 0.618·75-s + 1.17·77-s − 3.07·87-s + 0.618·89-s − 97-s + 1.00·99-s + ⋯ |
| L(s) = 1 | + 1.61·3-s − 1.17·5-s + 1.90·7-s + 1.61·9-s + 0.618·11-s − 1.90·15-s + 3.07·21-s − 1.17·23-s + 0.381·25-s + 27-s − 1.90·29-s + 1.00·33-s − 2.23·35-s + 1.90·37-s − 1.61·43-s − 1.90·45-s + 2.61·49-s − 0.726·55-s + 3.07·63-s − 1.90·69-s − 1.61·73-s + 0.618·75-s + 1.17·77-s − 3.07·87-s + 0.618·89-s − 97-s + 1.00·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.207806549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.207806549\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 - 1.61T + T^{2} \) |
| 5 | \( 1 + 1.17T + T^{2} \) |
| 7 | \( 1 - 1.90T + T^{2} \) |
| 11 | \( 1 - 0.618T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.17T + T^{2} \) |
| 29 | \( 1 + 1.90T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.90T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.61T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + 1.61T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.618T + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.638730004384222042260604560119, −8.036756275247097456620194786021, −7.76363768769541073236035885158, −7.13273792258905385610917589810, −5.76796274790677487224462920899, −4.58450125576986025423162466947, −4.10805507489027854891849994526, −3.44413790429507204961278484167, −2.23470792740488525045108630381, −1.51325256976533327581302914130,
1.51325256976533327581302914130, 2.23470792740488525045108630381, 3.44413790429507204961278484167, 4.10805507489027854891849994526, 4.58450125576986025423162466947, 5.76796274790677487224462920899, 7.13273792258905385610917589810, 7.76363768769541073236035885158, 8.036756275247097456620194786021, 8.638730004384222042260604560119
