| L(s) = 1 | − 2.45·2-s + 1.58·3-s + 4.05·4-s − 3.55·5-s − 3.90·6-s − 1.72·7-s − 5.04·8-s − 0.477·9-s + 8.73·10-s + 0.521·11-s + 6.43·12-s − 6.48·13-s + 4.24·14-s − 5.63·15-s + 4.30·16-s − 0.0823·17-s + 1.17·18-s − 1.33·19-s − 14.3·20-s − 2.74·21-s − 1.28·22-s + 1.60·23-s − 8.01·24-s + 7.61·25-s + 15.9·26-s − 5.52·27-s − 6.99·28-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 0.916·3-s + 2.02·4-s − 1.58·5-s − 1.59·6-s − 0.652·7-s − 1.78·8-s − 0.159·9-s + 2.76·10-s + 0.157·11-s + 1.85·12-s − 1.79·13-s + 1.13·14-s − 1.45·15-s + 1.07·16-s − 0.0199·17-s + 0.277·18-s − 0.305·19-s − 3.21·20-s − 0.598·21-s − 0.273·22-s + 0.335·23-s − 1.63·24-s + 1.52·25-s + 3.13·26-s − 1.06·27-s − 1.32·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08436350935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08436350935\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5077 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.45T + 2T^{2} \) |
| 3 | \( 1 - 1.58T + 3T^{2} \) |
| 5 | \( 1 + 3.55T + 5T^{2} \) |
| 7 | \( 1 + 1.72T + 7T^{2} \) |
| 11 | \( 1 - 0.521T + 11T^{2} \) |
| 13 | \( 1 + 6.48T + 13T^{2} \) |
| 17 | \( 1 + 0.0823T + 17T^{2} \) |
| 19 | \( 1 + 1.33T + 19T^{2} \) |
| 23 | \( 1 - 1.60T + 23T^{2} \) |
| 29 | \( 1 + 5.29T + 29T^{2} \) |
| 31 | \( 1 + 3.36T + 31T^{2} \) |
| 37 | \( 1 - 1.70T + 37T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 0.681T + 47T^{2} \) |
| 53 | \( 1 + 5.03T + 53T^{2} \) |
| 59 | \( 1 - 3.94T + 59T^{2} \) |
| 61 | \( 1 - 4.73T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 3.21T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 8.37T + 83T^{2} \) |
| 89 | \( 1 - 5.06T + 89T^{2} \) |
| 97 | \( 1 + 9.06T + 97T^{2} \) |
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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
−8.113992731577510558275389862433, −7.915446978229926872865696640792, −7.00615291994394067502165105598, −6.85869669058855305395261058452, −5.38309510587203931695005671105, −4.26920390487367769286386673385, −3.33296929733516363652261949484, −2.75712617516844992772617836128, −1.79469421421637012349279387168, −0.18433338596622281648753936448,
0.18433338596622281648753936448, 1.79469421421637012349279387168, 2.75712617516844992772617836128, 3.33296929733516363652261949484, 4.26920390487367769286386673385, 5.38309510587203931695005671105, 6.85869669058855305395261058452, 7.00615291994394067502165105598, 7.915446978229926872865696640792, 8.113992731577510558275389862433
