| L(s) = 1 | + 2·3-s + 4·5-s + 7-s + 9-s + 11-s − 13-s + 8·15-s + 2·17-s + 4·19-s + 2·21-s + 8·23-s + 11·25-s − 4·27-s − 10·29-s + 8·31-s + 2·33-s + 4·35-s + 2·37-s − 2·39-s − 10·41-s − 10·43-s + 4·45-s − 12·47-s + 49-s + 4·51-s − 10·53-s + 4·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 2.06·15-s + 0.485·17-s + 0.917·19-s + 0.436·21-s + 1.66·23-s + 11/5·25-s − 0.769·27-s − 1.85·29-s + 1.43·31-s + 0.348·33-s + 0.676·35-s + 0.328·37-s − 0.320·39-s − 1.56·41-s − 1.52·43-s + 0.596·45-s − 1.75·47-s + 1/7·49-s + 0.560·51-s − 1.37·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.505094635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.505094635\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.575172119398585934360496177317, −7.86392127858322128221830575282, −6.97363537531385285180830273371, −6.30620015862887518758120058348, −5.28687177524332969790499527324, −4.97769517672466023564189186272, −3.46951794795085193016933303588, −2.91598702828819945074221888314, −1.98833431488114325574814109799, −1.33064476050119544753558694072,
1.33064476050119544753558694072, 1.98833431488114325574814109799, 2.91598702828819945074221888314, 3.46951794795085193016933303588, 4.97769517672466023564189186272, 5.28687177524332969790499527324, 6.30620015862887518758120058348, 6.97363537531385285180830273371, 7.86392127858322128221830575282, 8.575172119398585934360496177317
