| L(s) = 1 | − 2-s + 4-s + 3.41·5-s − 8-s − 3.41·10-s − 11-s + 1.82·13-s + 16-s + 7.65·17-s − 3.41·19-s + 3.41·20-s + 22-s − 2.24·23-s + 6.65·25-s − 1.82·26-s + 8.65·29-s − 4·31-s − 32-s − 7.65·34-s − 6.58·37-s + 3.41·38-s − 3.41·40-s + 2.58·41-s + 5.65·43-s − 44-s + 2.24·46-s − 6.48·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.52·5-s − 0.353·8-s − 1.07·10-s − 0.301·11-s + 0.507·13-s + 0.250·16-s + 1.85·17-s − 0.783·19-s + 0.763·20-s + 0.213·22-s − 0.467·23-s + 1.33·25-s − 0.358·26-s + 1.60·29-s − 0.718·31-s − 0.176·32-s − 1.31·34-s − 1.08·37-s + 0.553·38-s − 0.539·40-s + 0.403·41-s + 0.862·43-s − 0.150·44-s + 0.330·46-s − 0.945·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.294929883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.294929883\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 3.41T + 5T^{2} \) |
| 13 | \( 1 - 1.82T + 13T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 19 | \( 1 + 3.41T + 19T^{2} \) |
| 23 | \( 1 + 2.24T + 23T^{2} \) |
| 29 | \( 1 - 8.65T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 6.58T + 37T^{2} \) |
| 41 | \( 1 - 2.58T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 + 6.48T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 8.41T + 59T^{2} \) |
| 61 | \( 1 - 6.17T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 3.07T + 71T^{2} \) |
| 73 | \( 1 + 6.58T + 73T^{2} \) |
| 79 | \( 1 + 4.75T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 - 4.48T + 89T^{2} \) |
| 97 | \( 1 - 1.82T + 97T^{2} \) |
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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
−7.76802327616644165772520562307, −6.94261568533896605644906872199, −6.36626956914172911989573151872, −5.58394062210325095176295216647, −5.37949360316654319972706474199, −4.13420906498941461097151240478, −3.16548352747667186226729363080, −2.40326634182346310482063540030, −1.64739423734673758886223729563, −0.842803186622131713617098643063,
0.842803186622131713617098643063, 1.64739423734673758886223729563, 2.40326634182346310482063540030, 3.16548352747667186226729363080, 4.13420906498941461097151240478, 5.37949360316654319972706474199, 5.58394062210325095176295216647, 6.36626956914172911989573151872, 6.94261568533896605644906872199, 7.76802327616644165772520562307
