| L(s) = 1 | + 3-s + 8·5-s − 17·7-s − 26·9-s + 70·11-s + 61·13-s + 8·15-s + 83·17-s − 19·19-s − 17·21-s + 115·23-s − 61·25-s − 53·27-s − 279·29-s − 72·31-s + 70·33-s − 136·35-s + 34·37-s + 61·39-s + 108·41-s + 192·43-s − 208·45-s − 392·47-s − 54·49-s + 83·51-s − 131·53-s + 560·55-s + ⋯ |
| L(s) = 1 | + 0.192·3-s + 0.715·5-s − 0.917·7-s − 0.962·9-s + 1.91·11-s + 1.30·13-s + 0.137·15-s + 1.18·17-s − 0.229·19-s − 0.176·21-s + 1.04·23-s − 0.487·25-s − 0.377·27-s − 1.78·29-s − 0.417·31-s + 0.369·33-s − 0.656·35-s + 0.151·37-s + 0.250·39-s + 0.411·41-s + 0.680·43-s − 0.689·45-s − 1.21·47-s − 0.157·49-s + 0.227·51-s − 0.339·53-s + 1.37·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.690172977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.690172977\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + p T \) |
| good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 5 | \( 1 - 8 T + p^{3} T^{2} \) |
| 7 | \( 1 + 17 T + p^{3} T^{2} \) |
| 11 | \( 1 - 70 T + p^{3} T^{2} \) |
| 13 | \( 1 - 61 T + p^{3} T^{2} \) |
| 17 | \( 1 - 83 T + p^{3} T^{2} \) |
| 23 | \( 1 - 5 p T + p^{3} T^{2} \) |
| 29 | \( 1 + 279 T + p^{3} T^{2} \) |
| 31 | \( 1 + 72 T + p^{3} T^{2} \) |
| 37 | \( 1 - 34 T + p^{3} T^{2} \) |
| 41 | \( 1 - 108 T + p^{3} T^{2} \) |
| 43 | \( 1 - 192 T + p^{3} T^{2} \) |
| 47 | \( 1 + 392 T + p^{3} T^{2} \) |
| 53 | \( 1 + 131 T + p^{3} T^{2} \) |
| 59 | \( 1 - 609 T + p^{3} T^{2} \) |
| 61 | \( 1 + 338 T + p^{3} T^{2} \) |
| 67 | \( 1 - 461 T + p^{3} T^{2} \) |
| 71 | \( 1 - 750 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1177 T + p^{3} T^{2} \) |
| 79 | \( 1 + 22 T + p^{3} T^{2} \) |
| 83 | \( 1 - 810 T + p^{3} T^{2} \) |
| 89 | \( 1 + 476 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1426 T + p^{3} T^{2} \) |
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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
−9.276233092838817232407378248251, −8.864140576354861380061253434329, −7.76607992946098285409730474520, −6.56817419297560162083123762057, −6.14067386863665930609530442811, −5.37888289565725550576091196400, −3.71819781474324443672830356922, −3.41039964425079254379869351903, −1.94755849058392082225080345299, −0.861851611062581382759056021152,
0.861851611062581382759056021152, 1.94755849058392082225080345299, 3.41039964425079254379869351903, 3.71819781474324443672830356922, 5.37888289565725550576091196400, 6.14067386863665930609530442811, 6.56817419297560162083123762057, 7.76607992946098285409730474520, 8.864140576354861380061253434329, 9.276233092838817232407378248251
