| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 2·10-s + 11-s + 2.58·13-s + 16-s − 2·17-s + 6.24·19-s − 2·20-s − 22-s − 0.828·23-s − 25-s − 2.58·26-s + 1.65·29-s − 2.24·31-s − 32-s + 2·34-s − 4.82·37-s − 6.24·38-s + 2·40-s − 0.343·41-s + 0.828·43-s + 44-s + 0.828·46-s − 11.8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s + 0.301·11-s + 0.717·13-s + 0.250·16-s − 0.485·17-s + 1.43·19-s − 0.447·20-s − 0.213·22-s − 0.172·23-s − 0.200·25-s − 0.507·26-s + 0.307·29-s − 0.402·31-s − 0.176·32-s + 0.342·34-s − 0.793·37-s − 1.01·38-s + 0.316·40-s − 0.0535·41-s + 0.126·43-s + 0.150·44-s + 0.122·46-s − 1.73·47-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2T + 5T^{2} \) |
| 13 | \( 1 - 2.58T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 6.24T + 19T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 29 | \( 1 - 1.65T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 + 4.82T + 37T^{2} \) |
| 41 | \( 1 + 0.343T + 41T^{2} \) |
| 43 | \( 1 - 0.828T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 6.48T + 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 - 5.41T + 61T^{2} \) |
| 67 | \( 1 + 6.82T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 0.343T + 73T^{2} \) |
| 79 | \( 1 - 0.485T + 79T^{2} \) |
| 83 | \( 1 - 9.07T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−7.46097777734601280385088201311, −6.82281213986347351775868191746, −6.16149469322688461261825954011, −5.32633217586220673048508144680, −4.50578519622270047206322049979, −3.60037246306004207729214130025, −3.17455506569084298960814445469, −1.96617408269351178127910299745, −1.08946590599935230753533481686, 0,
1.08946590599935230753533481686, 1.96617408269351178127910299745, 3.17455506569084298960814445469, 3.60037246306004207729214130025, 4.50578519622270047206322049979, 5.32633217586220673048508144680, 6.16149469322688461261825954011, 6.82281213986347351775868191746, 7.46097777734601280385088201311
