| L(s) = 1 | + 3·3-s + 5·5-s + 16·7-s + 9·9-s + 28·11-s − 26·13-s + 15·15-s − 62·17-s + 68·19-s + 48·21-s + 208·23-s + 25·25-s + 27·27-s − 58·29-s − 160·31-s + 84·33-s + 80·35-s + 270·37-s − 78·39-s + 282·41-s − 76·43-s + 45·45-s + 280·47-s − 87·49-s − 186·51-s − 210·53-s + 140·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.863·7-s + 1/3·9-s + 0.767·11-s − 0.554·13-s + 0.258·15-s − 0.884·17-s + 0.821·19-s + 0.498·21-s + 1.88·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.926·31-s + 0.443·33-s + 0.386·35-s + 1.19·37-s − 0.320·39-s + 1.07·41-s − 0.269·43-s + 0.149·45-s + 0.868·47-s − 0.253·49-s − 0.510·51-s − 0.544·53-s + 0.343·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.665140859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.665140859\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 62 T + p^{3} T^{2} \) |
| 19 | \( 1 - 68 T + p^{3} T^{2} \) |
| 23 | \( 1 - 208 T + p^{3} T^{2} \) |
| 29 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 160 T + p^{3} T^{2} \) |
| 37 | \( 1 - 270 T + p^{3} T^{2} \) |
| 41 | \( 1 - 282 T + p^{3} T^{2} \) |
| 43 | \( 1 + 76 T + p^{3} T^{2} \) |
| 47 | \( 1 - 280 T + p^{3} T^{2} \) |
| 53 | \( 1 + 210 T + p^{3} T^{2} \) |
| 59 | \( 1 + 196 T + p^{3} T^{2} \) |
| 61 | \( 1 - 742 T + p^{3} T^{2} \) |
| 67 | \( 1 + 836 T + p^{3} T^{2} \) |
| 71 | \( 1 - 504 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1062 T + p^{3} T^{2} \) |
| 79 | \( 1 + 768 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1052 T + p^{3} T^{2} \) |
| 89 | \( 1 + 726 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1406 T + p^{3} T^{2} \) |
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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
−11.56134148656735557168942010333, −10.82165315964331447991349244707, −9.463433671064582827237795677151, −8.958853374898012048455547699891, −7.71299994166706678632129253201, −6.81558311800312019467515407985, −5.36293781965134060438391282552, −4.27287616417712267745955780791, −2.72404022815463313396267882602, −1.36261459126261242531345775005,
1.36261459126261242531345775005, 2.72404022815463313396267882602, 4.27287616417712267745955780791, 5.36293781965134060438391282552, 6.81558311800312019467515407985, 7.71299994166706678632129253201, 8.958853374898012048455547699891, 9.463433671064582827237795677151, 10.82165315964331447991349244707, 11.56134148656735557168942010333
