| L(s) = 1 | − 2-s − 3-s + 4-s + 3·5-s + 6-s + 7-s − 8-s − 2·9-s − 3·10-s − 6·11-s − 12-s + 13-s − 14-s − 3·15-s + 16-s + 2·18-s + 2·19-s + 3·20-s − 21-s + 6·22-s + 24-s + 4·25-s − 26-s + 5·27-s + 28-s − 6·29-s + 3·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.948·10-s − 1.80·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.471·18-s + 0.458·19-s + 0.670·20-s − 0.218·21-s + 1.27·22-s + 0.204·24-s + 4/5·25-s − 0.196·26-s + 0.962·27-s + 0.188·28-s − 1.11·29-s + 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7514 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7514 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.125411348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.125411348\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.021906580411022736283755507389, −7.27359556589843556613658536576, −6.34677629131558710598757660896, −5.77137129300962012745774581204, −5.38018266196028383329603462561, −4.65668153893619531664347779146, −3.15617299774676756464792678109, −2.50470628431012273081407899454, −1.75576484932556874056389447796, −0.59657491686844749103695751194,
0.59657491686844749103695751194, 1.75576484932556874056389447796, 2.50470628431012273081407899454, 3.15617299774676756464792678109, 4.65668153893619531664347779146, 5.38018266196028383329603462561, 5.77137129300962012745774581204, 6.34677629131558710598757660896, 7.27359556589843556613658536576, 8.021906580411022736283755507389
