| L(s) = 1 | − 2-s + 4-s + 2.44·5-s − 4.35·7-s − 8-s − 2.44·10-s − 6.34·11-s − 5.15·13-s + 4.35·14-s + 16-s + 6.82·17-s − 5.18·19-s + 2.44·20-s + 6.34·22-s + 5.35·23-s + 0.978·25-s + 5.15·26-s − 4.35·28-s − 1.96·29-s − 0.198·31-s − 32-s − 6.82·34-s − 10.6·35-s + 3.63·37-s + 5.18·38-s − 2.44·40-s − 10.2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.09·5-s − 1.64·7-s − 0.353·8-s − 0.773·10-s − 1.91·11-s − 1.43·13-s + 1.16·14-s + 0.250·16-s + 1.65·17-s − 1.18·19-s + 0.546·20-s + 1.35·22-s + 1.11·23-s + 0.195·25-s + 1.01·26-s − 0.823·28-s − 0.364·29-s − 0.0355·31-s − 0.176·32-s − 1.17·34-s − 1.80·35-s + 0.596·37-s + 0.841·38-s − 0.386·40-s − 1.60·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7657463240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7657463240\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 - 2.44T + 5T^{2} \) |
| 7 | \( 1 + 4.35T + 7T^{2} \) |
| 11 | \( 1 + 6.34T + 11T^{2} \) |
| 13 | \( 1 + 5.15T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 + 5.18T + 19T^{2} \) |
| 23 | \( 1 - 5.35T + 23T^{2} \) |
| 29 | \( 1 + 1.96T + 29T^{2} \) |
| 31 | \( 1 + 0.198T + 31T^{2} \) |
| 37 | \( 1 - 3.63T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 3.69T + 43T^{2} \) |
| 47 | \( 1 - 7.25T + 47T^{2} \) |
| 53 | \( 1 - 5.62T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 + 5.03T + 67T^{2} \) |
| 71 | \( 1 - 6.16T + 71T^{2} \) |
| 73 | \( 1 - 8.65T + 73T^{2} \) |
| 79 | \( 1 - 4.78T + 79T^{2} \) |
| 83 | \( 1 + 0.911T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 9.50T + 97T^{2} \) |
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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
−8.523411644445286324030220027055, −7.65242727086508870444625723225, −7.07553812234364556852107686698, −6.30396848830301957270219082358, −5.53226776776243258884962595138, −5.06025494586985465400127532005, −3.46316799613955853481410271575, −2.69000167545766705950058038907, −2.16715669669732196541148813706, −0.51659140580223560724074911490,
0.51659140580223560724074911490, 2.16715669669732196541148813706, 2.69000167545766705950058038907, 3.46316799613955853481410271575, 5.06025494586985465400127532005, 5.53226776776243258884962595138, 6.30396848830301957270219082358, 7.07553812234364556852107686698, 7.65242727086508870444625723225, 8.523411644445286324030220027055
