L(s) = 1 | + 3-s − 2.54·5-s − 0.525·7-s + 9-s − 5.59·11-s + 1.57·13-s − 2.54·15-s + 4.20·17-s − 3.78·19-s − 0.525·21-s + 2.45·23-s + 1.47·25-s + 27-s − 4.24·29-s + 5.76·31-s − 5.59·33-s + 1.33·35-s − 2.67·37-s + 1.57·39-s + 10.7·41-s − 8.36·43-s − 2.54·45-s − 4.77·47-s − 6.72·49-s + 4.20·51-s − 2.34·53-s + 14.2·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.13·5-s − 0.198·7-s + 0.333·9-s − 1.68·11-s + 0.437·13-s − 0.656·15-s + 1.02·17-s − 0.869·19-s − 0.114·21-s + 0.511·23-s + 0.294·25-s + 0.192·27-s − 0.787·29-s + 1.03·31-s − 0.973·33-s + 0.225·35-s − 0.439·37-s + 0.252·39-s + 1.68·41-s − 1.27·43-s − 0.379·45-s − 0.697·47-s − 0.960·49-s + 0.589·51-s − 0.322·53-s + 1.91·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.302237262\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.302237262\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 + 2.54T + 5T^{2} \) |
| 7 | \( 1 + 0.525T + 7T^{2} \) |
| 11 | \( 1 + 5.59T + 11T^{2} \) |
| 13 | \( 1 - 1.57T + 13T^{2} \) |
| 17 | \( 1 - 4.20T + 17T^{2} \) |
| 19 | \( 1 + 3.78T + 19T^{2} \) |
| 23 | \( 1 - 2.45T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - 5.76T + 31T^{2} \) |
| 37 | \( 1 + 2.67T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 8.36T + 43T^{2} \) |
| 47 | \( 1 + 4.77T + 47T^{2} \) |
| 53 | \( 1 + 2.34T + 53T^{2} \) |
| 59 | \( 1 - 5.25T + 59T^{2} \) |
| 61 | \( 1 + 5.02T + 61T^{2} \) |
| 67 | \( 1 + 8.91T + 67T^{2} \) |
| 71 | \( 1 - 5.10T + 71T^{2} \) |
| 73 | \( 1 + 2.16T + 73T^{2} \) |
| 79 | \( 1 - 2.52T + 79T^{2} \) |
| 83 | \( 1 - 4.01T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−7.977476341858645065112061466363, −7.41233647358070559119754097774, −6.57476085809938451736267980230, −5.70613200287024004362306585942, −4.90934420841920302913566478226, −4.22695995003818990183572963076, −3.35271858476553575864245098324, −2.92849046126144165695046199996, −1.87479976127938756280110250011, −0.53048879360008684140098530121,
0.53048879360008684140098530121, 1.87479976127938756280110250011, 2.92849046126144165695046199996, 3.35271858476553575864245098324, 4.22695995003818990183572963076, 4.90934420841920302913566478226, 5.70613200287024004362306585942, 6.57476085809938451736267980230, 7.41233647358070559119754097774, 7.977476341858645065112061466363