L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.541 + 1.30i)3-s − 1.00i·4-s + (−0.541 − 1.30i)6-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.541 − 1.30i)11-s + (1.30 + 0.541i)12-s − 1.00·16-s + 1.00·18-s + (1.41 − 1.41i)19-s + (1.30 + 0.541i)22-s + (−1.30 + 0.541i)24-s + (0.707 + 0.707i)25-s + (0.707 − 0.707i)32-s + 2·33-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.541 + 1.30i)3-s − 1.00i·4-s + (−0.541 − 1.30i)6-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.541 − 1.30i)11-s + (1.30 + 0.541i)12-s − 1.00·16-s + 1.00·18-s + (1.41 − 1.41i)19-s + (1.30 + 0.541i)22-s + (−1.30 + 0.541i)24-s + (0.707 + 0.707i)25-s + (0.707 − 0.707i)32-s + 2·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6600856743\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6600856743\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \) |
| 5 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 7 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 23 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 37 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2} \) |
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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
−9.208758044873405794211381027095, −8.856547106993625580926671051434, −7.82540803307037307410419426221, −7.07467454973755129369752513671, −6.07628795794821602788289840810, −5.30161669880375854172775732101, −4.98698810135584400079064020631, −3.79128197965685282396469971070, −2.72545579043543608511953858332, −0.799448913768434944843174636956,
1.03999279212941085974636563152, 1.92884195425825490444998231441, 2.82491437746651763237380595532, 4.06915402546625113136897137738, 5.12972694795297138449187195287, 6.16001193880631556945637712567, 6.99796796362960543429922071283, 7.67986060992445803472892810839, 7.955969522727569088833346740490, 9.142128715061526762740533705160