| L(s) = 1 | − 2.41·2-s + 3.82·4-s + 3.41·7-s − 4.41·8-s − 2.82·11-s − 8.24·14-s + 2.99·16-s − 7.41·17-s − 6.24·19-s + 6.82·22-s − 23-s + 13.0·28-s + 8.48·29-s + 8.48·31-s + 1.58·32-s + 17.8·34-s + 4.82·37-s + 15.0·38-s − 1.65·41-s + 1.75·43-s − 10.8·44-s + 2.41·46-s + 0.343·47-s + 4.65·49-s + 5.07·53-s − 15.0·56-s − 20.4·58-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 1.91·4-s + 1.29·7-s − 1.56·8-s − 0.852·11-s − 2.20·14-s + 0.749·16-s − 1.79·17-s − 1.43·19-s + 1.45·22-s − 0.208·23-s + 2.47·28-s + 1.57·29-s + 1.52·31-s + 0.280·32-s + 3.06·34-s + 0.793·37-s + 2.44·38-s − 0.258·41-s + 0.267·43-s − 1.63·44-s + 0.355·46-s + 0.0500·47-s + 0.665·49-s + 0.696·53-s − 2.01·56-s − 2.68·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 7.41T + 17T^{2} \) |
| 19 | \( 1 + 6.24T + 19T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 4.82T + 37T^{2} \) |
| 41 | \( 1 + 1.65T + 41T^{2} \) |
| 43 | \( 1 - 1.75T + 43T^{2} \) |
| 47 | \( 1 - 0.343T + 47T^{2} \) |
| 53 | \( 1 - 5.07T + 53T^{2} \) |
| 59 | \( 1 - 7.65T + 59T^{2} \) |
| 61 | \( 1 + 0.828T + 61T^{2} \) |
| 67 | \( 1 + 8.58T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 - 7.89T + 79T^{2} \) |
| 83 | \( 1 + 6.82T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.117403323211770402038231137101, −7.44212969276821585006702069987, −6.64437448599803901801705901421, −6.06218994199601855879909151256, −4.71959930178482882515599064814, −4.41159938244136802471650755422, −2.60747539863987050435024064441, −2.21744297483320697043169794135, −1.17652089248666232955865544009, 0,
1.17652089248666232955865544009, 2.21744297483320697043169794135, 2.60747539863987050435024064441, 4.41159938244136802471650755422, 4.71959930178482882515599064814, 6.06218994199601855879909151256, 6.64437448599803901801705901421, 7.44212969276821585006702069987, 8.117403323211770402038231137101
