| L(s) = 1 | − 2-s − 3-s + 4-s + 3.33·5-s + 6-s − 4.38·7-s − 8-s + 9-s − 3.33·10-s + 5.67·11-s − 12-s − 2.77·13-s + 4.38·14-s − 3.33·15-s + 16-s − 3.53·17-s − 18-s + 3.33·20-s + 4.38·21-s − 5.67·22-s + 4·23-s + 24-s + 6.09·25-s + 2.77·26-s − 27-s − 4.38·28-s + 6.80·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.48·5-s + 0.408·6-s − 1.65·7-s − 0.353·8-s + 0.333·9-s − 1.05·10-s + 1.71·11-s − 0.288·12-s − 0.768·13-s + 1.17·14-s − 0.860·15-s + 0.250·16-s − 0.857·17-s − 0.235·18-s + 0.744·20-s + 0.957·21-s − 1.21·22-s + 0.834·23-s + 0.204·24-s + 1.21·25-s + 0.543·26-s − 0.192·27-s − 0.829·28-s + 1.26·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.157376154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.157376154\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 3.33T + 5T^{2} \) |
| 7 | \( 1 + 4.38T + 7T^{2} \) |
| 11 | \( 1 - 5.67T + 11T^{2} \) |
| 13 | \( 1 + 2.77T + 13T^{2} \) |
| 17 | \( 1 + 3.53T + 17T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 - 4.17T + 37T^{2} \) |
| 41 | \( 1 - 4.95T + 41T^{2} \) |
| 43 | \( 1 + 7.89T + 43T^{2} \) |
| 47 | \( 1 - 5.42T + 47T^{2} \) |
| 53 | \( 1 - 4.80T + 53T^{2} \) |
| 59 | \( 1 - 0.795T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 - 6.47T + 67T^{2} \) |
| 71 | \( 1 + 3.89T + 71T^{2} \) |
| 73 | \( 1 - 4.21T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 1.03T + 83T^{2} \) |
| 89 | \( 1 - 2.48T + 89T^{2} \) |
| 97 | \( 1 + 1.24T + 97T^{2} \) |
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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
−9.245863817330498811172294915507, −8.757286773345230426641013180168, −7.07702680704358531648283160710, −6.71732633788386924295113927390, −6.19079948618383445634328234028, −5.42870873465157731488678190471, −4.17484984237605399838521847763, −2.98629666951075489333798828764, −2.02025929467579279099980574054, −0.808407138222130960920237068758,
0.808407138222130960920237068758, 2.02025929467579279099980574054, 2.98629666951075489333798828764, 4.17484984237605399838521847763, 5.42870873465157731488678190471, 6.19079948618383445634328234028, 6.71732633788386924295113927390, 7.07702680704358531648283160710, 8.757286773345230426641013180168, 9.245863817330498811172294915507
