L(s) = 1 | − 3.41·3-s + 0.828·7-s + 8.65·9-s + 2·11-s + 6.24·13-s − 0.828·17-s + 19-s − 2.82·21-s − 6·23-s − 19.3·27-s − 6.48·29-s + 6.82·31-s − 6.82·33-s + 1.75·37-s − 21.3·39-s + 3.65·41-s + 4.82·43-s − 4.82·47-s − 6.31·49-s + 2.82·51-s − 9.07·53-s − 3.41·57-s − 13.6·59-s − 13.6·61-s + 7.17·63-s − 3.41·67-s + 20.4·69-s + ⋯ |
L(s) = 1 | − 1.97·3-s + 0.313·7-s + 2.88·9-s + 0.603·11-s + 1.73·13-s − 0.200·17-s + 0.229·19-s − 0.617·21-s − 1.25·23-s − 3.71·27-s − 1.20·29-s + 1.22·31-s − 1.18·33-s + 0.288·37-s − 3.41·39-s + 0.571·41-s + 0.736·43-s − 0.704·47-s − 0.901·49-s + 0.396·51-s − 1.24·53-s − 0.452·57-s − 1.77·59-s − 1.74·61-s + 0.903·63-s − 0.417·67-s + 2.46·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 3.41T + 3T^{2} \) |
| 7 | \( 1 - 0.828T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 6.24T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 6.48T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 - 1.75T + 37T^{2} \) |
| 41 | \( 1 - 3.65T + 41T^{2} \) |
| 43 | \( 1 - 4.82T + 43T^{2} \) |
| 47 | \( 1 + 4.82T + 47T^{2} \) |
| 53 | \( 1 + 9.07T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 + 3.41T + 67T^{2} \) |
| 71 | \( 1 + 5.17T + 71T^{2} \) |
| 73 | \( 1 - 2.48T + 73T^{2} \) |
| 79 | \( 1 + 1.65T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 6.48T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−7.43960140065000642912778837680, −6.40097313956949129748397024172, −6.17512027789531467168869845890, −5.68859993282908055755918509507, −4.64687975272024600952456563725, −4.27312918786731683231944592029, −3.40709066735849313178600114329, −1.70572480122082180628276553013, −1.19998502955432035098250849677, 0,
1.19998502955432035098250849677, 1.70572480122082180628276553013, 3.40709066735849313178600114329, 4.27312918786731683231944592029, 4.64687975272024600952456563725, 5.68859993282908055755918509507, 6.17512027789531467168869845890, 6.40097313956949129748397024172, 7.43960140065000642912778837680