| L(s) = 1 | + 5·2-s + 4·3-s + 17·4-s + 5·5-s + 20·6-s − 32·7-s + 45·8-s − 11·9-s + 25·10-s − 12·11-s + 68·12-s − 42·13-s − 160·14-s + 20·15-s + 89·16-s + 114·17-s − 55·18-s + 19·19-s + 85·20-s − 128·21-s − 60·22-s + 160·23-s + 180·24-s + 25·25-s − 210·26-s − 152·27-s − 544·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 0.769·3-s + 17/8·4-s + 0.447·5-s + 1.36·6-s − 1.72·7-s + 1.98·8-s − 0.407·9-s + 0.790·10-s − 0.328·11-s + 1.63·12-s − 0.896·13-s − 3.05·14-s + 0.344·15-s + 1.39·16-s + 1.62·17-s − 0.720·18-s + 0.229·19-s + 0.950·20-s − 1.33·21-s − 0.581·22-s + 1.45·23-s + 1.53·24-s + 1/5·25-s − 1.58·26-s − 1.08·27-s − 3.67·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.408321459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.408321459\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p T \) |
| 19 | \( 1 - p T \) |
| good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 42 T + p^{3} T^{2} \) |
| 17 | \( 1 - 114 T + p^{3} T^{2} \) |
| 23 | \( 1 - 160 T + p^{3} T^{2} \) |
| 29 | \( 1 - 214 T + p^{3} T^{2} \) |
| 31 | \( 1 + 144 T + p^{3} T^{2} \) |
| 37 | \( 1 - 94 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 308 T + p^{3} T^{2} \) |
| 47 | \( 1 - 184 T + p^{3} T^{2} \) |
| 53 | \( 1 + 274 T + p^{3} T^{2} \) |
| 59 | \( 1 - 276 T + p^{3} T^{2} \) |
| 61 | \( 1 + 826 T + p^{3} T^{2} \) |
| 67 | \( 1 - 52 T + p^{3} T^{2} \) |
| 71 | \( 1 + 344 T + p^{3} T^{2} \) |
| 73 | \( 1 + 166 T + p^{3} T^{2} \) |
| 79 | \( 1 + 688 T + p^{3} T^{2} \) |
| 83 | \( 1 - 12 p T + p^{3} T^{2} \) |
| 89 | \( 1 - 1578 T + p^{3} T^{2} \) |
| 97 | \( 1 - 786 T + p^{3} T^{2} \) |
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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
−13.49680944465533968447342213490, −12.78853986612772899786442431254, −11.97414837086227405111099186491, −10.34014236365266079026022258191, −9.275731422762311996957323002767, −7.41037064926758717373541794522, −6.24903790107919138185851345463, −5.18483478460953901704771831976, −3.33369221002320210030548149097, −2.78157404566324258854931061713,
2.78157404566324258854931061713, 3.33369221002320210030548149097, 5.18483478460953901704771831976, 6.24903790107919138185851345463, 7.41037064926758717373541794522, 9.275731422762311996957323002767, 10.34014236365266079026022258191, 11.97414837086227405111099186491, 12.78853986612772899786442431254, 13.49680944465533968447342213490
