| L(s) = 1 | + 2.06·2-s − 0.977·3-s + 2.27·4-s − 2.02·6-s + 1.48·7-s + 0.570·8-s − 2.04·9-s + 3.74·11-s − 2.22·12-s + 0.0515·13-s + 3.07·14-s − 3.37·16-s + 6.83·17-s − 4.22·18-s − 3.63·19-s − 1.45·21-s + 7.74·22-s − 0.204·23-s − 0.557·24-s + 0.106·26-s + 4.93·27-s + 3.38·28-s + 2.87·29-s + 9.41·31-s − 8.11·32-s − 3.66·33-s + 14.1·34-s + ⋯ |
| L(s) = 1 | + 1.46·2-s − 0.564·3-s + 1.13·4-s − 0.825·6-s + 0.562·7-s + 0.201·8-s − 0.681·9-s + 1.12·11-s − 0.642·12-s + 0.0142·13-s + 0.821·14-s − 0.843·16-s + 1.65·17-s − 0.996·18-s − 0.833·19-s − 0.317·21-s + 1.65·22-s − 0.0425·23-s − 0.113·24-s + 0.0208·26-s + 0.948·27-s + 0.639·28-s + 0.533·29-s + 1.69·31-s − 1.43·32-s − 0.637·33-s + 2.42·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.955568141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.955568141\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 2.06T + 2T^{2} \) |
| 3 | \( 1 + 0.977T + 3T^{2} \) |
| 7 | \( 1 - 1.48T + 7T^{2} \) |
| 11 | \( 1 - 3.74T + 11T^{2} \) |
| 13 | \( 1 - 0.0515T + 13T^{2} \) |
| 17 | \( 1 - 6.83T + 17T^{2} \) |
| 19 | \( 1 + 3.63T + 19T^{2} \) |
| 23 | \( 1 + 0.204T + 23T^{2} \) |
| 29 | \( 1 - 2.87T + 29T^{2} \) |
| 31 | \( 1 - 9.41T + 31T^{2} \) |
| 37 | \( 1 + 9.14T + 37T^{2} \) |
| 41 | \( 1 - 4.67T + 41T^{2} \) |
| 43 | \( 1 + 2.78T + 43T^{2} \) |
| 47 | \( 1 - 1.18T + 47T^{2} \) |
| 53 | \( 1 + 4.30T + 53T^{2} \) |
| 59 | \( 1 + 4.09T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 9.06T + 67T^{2} \) |
| 71 | \( 1 - 4.41T + 71T^{2} \) |
| 73 | \( 1 - 8.80T + 73T^{2} \) |
| 79 | \( 1 + 7.87T + 79T^{2} \) |
| 83 | \( 1 - 2.66T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−8.086428701098662053773207997642, −6.94797744851390405457539031142, −6.36960384748349501333230742895, −5.85751492932783211758711724770, −5.09530787639303512078826790956, −4.63276293613398557680513563296, −3.71681864898618591632057347883, −3.12917084065072235437399082228, −2.08220343493636846885032354827, −0.879349200945060192226435943120,
0.879349200945060192226435943120, 2.08220343493636846885032354827, 3.12917084065072235437399082228, 3.71681864898618591632057347883, 4.63276293613398557680513563296, 5.09530787639303512078826790956, 5.85751492932783211758711724770, 6.36960384748349501333230742895, 6.94797744851390405457539031142, 8.086428701098662053773207997642
