| L(s) = 1 | + 1.10·2-s − 0.771·4-s − 2.10·7-s − 3.07·8-s − 1.47·11-s + 3.47·13-s − 2.32·14-s − 1.86·16-s − 5.48·17-s − 2.79·19-s − 1.63·22-s + 3.87·23-s + 3.85·26-s + 1.61·28-s + 2.76·29-s − 10.5·31-s + 4.07·32-s − 6.07·34-s − 37-s − 3.09·38-s + 2.73·41-s + 0.688·43-s + 1.13·44-s + 4.29·46-s + 4.12·47-s − 2.58·49-s − 2.67·52-s + ⋯ |
| L(s) = 1 | + 0.783·2-s − 0.385·4-s − 0.793·7-s − 1.08·8-s − 0.445·11-s + 0.963·13-s − 0.622·14-s − 0.465·16-s − 1.33·17-s − 0.641·19-s − 0.349·22-s + 0.807·23-s + 0.755·26-s + 0.306·28-s + 0.513·29-s − 1.90·31-s + 0.720·32-s − 1.04·34-s − 0.164·37-s − 0.502·38-s + 0.427·41-s + 0.104·43-s + 0.171·44-s + 0.632·46-s + 0.600·47-s − 0.369·49-s − 0.371·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.380046392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.380046392\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 1.10T + 2T^{2} \) |
| 7 | \( 1 + 2.10T + 7T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 + 5.48T + 17T^{2} \) |
| 19 | \( 1 + 2.79T + 19T^{2} \) |
| 23 | \( 1 - 3.87T + 23T^{2} \) |
| 29 | \( 1 - 2.76T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 43 | \( 1 - 0.688T + 43T^{2} \) |
| 47 | \( 1 - 4.12T + 47T^{2} \) |
| 53 | \( 1 + 9.67T + 53T^{2} \) |
| 59 | \( 1 - 9.61T + 59T^{2} \) |
| 61 | \( 1 + 7.54T + 61T^{2} \) |
| 67 | \( 1 + 5.95T + 67T^{2} \) |
| 71 | \( 1 - 2.66T + 71T^{2} \) |
| 73 | \( 1 + 9.13T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 8.88T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−7.78393923670322186340600505340, −6.82250625333228054211766564843, −6.33546109457460564626420872858, −5.70827468545738657326781384566, −4.93481562671143541280363302790, −4.26871706780659804782093552494, −3.56339341431338388486832733897, −2.95070744048078442920652404619, −1.96576534992836944839991058189, −0.49157498891012072647486336674,
0.49157498891012072647486336674, 1.96576534992836944839991058189, 2.95070744048078442920652404619, 3.56339341431338388486832733897, 4.26871706780659804782093552494, 4.93481562671143541280363302790, 5.70827468545738657326781384566, 6.33546109457460564626420872858, 6.82250625333228054211766564843, 7.78393923670322186340600505340
