L(s) = 1 | − 3-s + 5-s − 3.37·7-s + 9-s − 0.627·11-s − 2·13-s − 15-s − 4.74·17-s + 0.627·19-s + 3.37·21-s − 3.37·23-s + 25-s − 27-s − 8.74·29-s + 31-s + 0.627·33-s − 3.37·35-s − 0.744·37-s + 2·39-s + 0.744·41-s − 0.627·43-s + 45-s + 6.74·47-s + 4.37·49-s + 4.74·51-s + 1.37·53-s − 0.627·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.27·7-s + 0.333·9-s − 0.189·11-s − 0.554·13-s − 0.258·15-s − 1.15·17-s + 0.144·19-s + 0.735·21-s − 0.703·23-s + 0.200·25-s − 0.192·27-s − 1.62·29-s + 0.179·31-s + 0.109·33-s − 0.570·35-s − 0.122·37-s + 0.320·39-s + 0.116·41-s − 0.0957·43-s + 0.149·45-s + 0.983·47-s + 0.624·49-s + 0.664·51-s + 0.188·53-s − 0.0846·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7418154829\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7418154829\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 + 0.627T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 4.74T + 17T^{2} \) |
| 19 | \( 1 - 0.627T + 19T^{2} \) |
| 23 | \( 1 + 3.37T + 23T^{2} \) |
| 29 | \( 1 + 8.74T + 29T^{2} \) |
| 37 | \( 1 + 0.744T + 37T^{2} \) |
| 41 | \( 1 - 0.744T + 41T^{2} \) |
| 43 | \( 1 + 0.627T + 43T^{2} \) |
| 47 | \( 1 - 6.74T + 47T^{2} \) |
| 53 | \( 1 - 1.37T + 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 3.37T + 71T^{2} \) |
| 73 | \( 1 + 8.11T + 73T^{2} \) |
| 79 | \( 1 - 4.62T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 1.37T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−7.66783905949348336052332401885, −7.10456628294832911001876840562, −6.40524094479749913293433631863, −5.91031111629671526535015958420, −5.21632208427655927573190629761, −4.34028292660842255750717023413, −3.59110078042870576215247492418, −2.63040779039133934182153000648, −1.86643750655640715952638526395, −0.42322978494496860810746944405,
0.42322978494496860810746944405, 1.86643750655640715952638526395, 2.63040779039133934182153000648, 3.59110078042870576215247492418, 4.34028292660842255750717023413, 5.21632208427655927573190629761, 5.91031111629671526535015958420, 6.40524094479749913293433631863, 7.10456628294832911001876840562, 7.66783905949348336052332401885