L(s) = 1 | + 3-s + 5-s + 3.82·7-s + 9-s − 1.41·11-s + 3.41·13-s + 15-s + 0.414·17-s + 4.58·19-s + 3.82·21-s + 23-s + 25-s + 27-s − 6.41·29-s + 31-s − 1.41·33-s + 3.82·35-s + 9.48·37-s + 3.41·39-s − 1.24·41-s − 3.65·43-s + 45-s − 2.24·47-s + 7.65·49-s + 0.414·51-s − 3.24·53-s − 1.41·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.44·7-s + 0.333·9-s − 0.426·11-s + 0.946·13-s + 0.258·15-s + 0.100·17-s + 1.05·19-s + 0.835·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s − 1.19·29-s + 0.179·31-s − 0.246·33-s + 0.647·35-s + 1.55·37-s + 0.546·39-s − 0.194·41-s − 0.557·43-s + 0.149·45-s − 0.327·47-s + 1.09·49-s + 0.0580·51-s − 0.445·53-s − 0.190·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.639019673\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.639019673\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 - 3.82T + 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 - 0.414T + 17T^{2} \) |
| 19 | \( 1 - 4.58T + 19T^{2} \) |
| 29 | \( 1 + 6.41T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 9.48T + 37T^{2} \) |
| 41 | \( 1 + 1.24T + 41T^{2} \) |
| 43 | \( 1 + 3.65T + 43T^{2} \) |
| 47 | \( 1 + 2.24T + 47T^{2} \) |
| 53 | \( 1 + 3.24T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 + 1.48T + 67T^{2} \) |
| 71 | \( 1 + 0.0710T + 71T^{2} \) |
| 73 | \( 1 + 6.24T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.104975851131140692663251958422, −7.67038220229458437581256168661, −6.86284105408132507350035227485, −5.84447752046096597333364242467, −5.27606114813668981785376092263, −4.51658444697396938229746598583, −3.66190094936129529957273659331, −2.74679469783100957452263129259, −1.82172341760027644092478262511, −1.10247808635851129708709328847,
1.10247808635851129708709328847, 1.82172341760027644092478262511, 2.74679469783100957452263129259, 3.66190094936129529957273659331, 4.51658444697396938229746598583, 5.27606114813668981785376092263, 5.84447752046096597333364242467, 6.86284105408132507350035227485, 7.67038220229458437581256168661, 8.104975851131140692663251958422