| L(s) = 1 | + 2.72·2-s + 5.41·4-s − 3.82·7-s + 9.30·8-s + 4.41·11-s + 3.67·13-s − 10.4·14-s + 14.5·16-s − 2.28·17-s − 2.39·19-s + 12.0·22-s − 0.265·23-s + 10.0·26-s − 20.7·28-s + 6.58·29-s + 2.34·31-s + 20.9·32-s − 6.22·34-s − 37-s − 6.51·38-s + 4.41·41-s − 7.71·43-s + 23.9·44-s − 0.722·46-s + 10.9·47-s + 7.64·49-s + 19.9·52-s + ⋯ |
| L(s) = 1 | + 1.92·2-s + 2.70·4-s − 1.44·7-s + 3.29·8-s + 1.33·11-s + 1.01·13-s − 2.78·14-s + 3.62·16-s − 0.554·17-s − 0.548·19-s + 2.56·22-s − 0.0553·23-s + 1.96·26-s − 3.91·28-s + 1.22·29-s + 0.420·31-s + 3.69·32-s − 1.06·34-s − 0.164·37-s − 1.05·38-s + 0.689·41-s − 1.17·43-s + 3.60·44-s − 0.106·46-s + 1.59·47-s + 1.09·49-s + 2.76·52-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
| \(L(1)\) |
\(\approx\) |
\(7.854658743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.854658743\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.72T + 2T^{2} \) |
| 7 | \( 1 + 3.82T + 7T^{2} \) |
| 11 | \( 1 - 4.41T + 11T^{2} \) |
| 13 | \( 1 - 3.67T + 13T^{2} \) |
| 17 | \( 1 + 2.28T + 17T^{2} \) |
| 19 | \( 1 + 2.39T + 19T^{2} \) |
| 23 | \( 1 + 0.265T + 23T^{2} \) |
| 29 | \( 1 - 6.58T + 29T^{2} \) |
| 31 | \( 1 - 2.34T + 31T^{2} \) |
| 41 | \( 1 - 4.41T + 41T^{2} \) |
| 43 | \( 1 + 7.71T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 0.109T + 53T^{2} \) |
| 59 | \( 1 - 2.00T + 59T^{2} \) |
| 61 | \( 1 - 3.96T + 61T^{2} \) |
| 67 | \( 1 + 6.80T + 67T^{2} \) |
| 71 | \( 1 - 5.79T + 71T^{2} \) |
| 73 | \( 1 - 0.140T + 73T^{2} \) |
| 79 | \( 1 + 6.62T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 8.94T + 97T^{2} \) |
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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
−7.28085569360308410370742502206, −6.67858485903302379599810448917, −6.27326159173880626533389050220, −5.92988521950616841312941783480, −4.86266575687214991371105800670, −4.11047513533952561135492240566, −3.68383533910178796064947035913, −3.01143986342094581768064474044, −2.19305317369170857090202724543, −1.07543945876769338540904342563,
1.07543945876769338540904342563, 2.19305317369170857090202724543, 3.01143986342094581768064474044, 3.68383533910178796064947035913, 4.11047513533952561135492240566, 4.86266575687214991371105800670, 5.92988521950616841312941783480, 6.27326159173880626533389050220, 6.67858485903302379599810448917, 7.28085569360308410370742502206
