| 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
−7.43077296037992944101561196873, −6.61169619931255753045439163928, −6.14474661702482663474943246708, −5.45066164102003787373877681616, −4.61370304557787753222486385129, −3.76369975183680166369929419344, −2.69109314124282389346293847004, −2.10715122356964009386802956626, −1.26760130220323677632751335347, 0,
1.26760130220323677632751335347, 2.10715122356964009386802956626, 2.69109314124282389346293847004, 3.76369975183680166369929419344, 4.61370304557787753222486385129, 5.45066164102003787373877681616, 6.14474661702482663474943246708, 6.61169619931255753045439163928, 7.43077296037992944101561196873
