| L(s) = 1 | + i·3-s + 4.43i·7-s − 9-s + 4.17·11-s − 4.43i·13-s − 1.35i·17-s − 2.70·19-s − 4.43·21-s + 1.35i·23-s − i·27-s + 7.25·29-s − 31-s + 4.17i·33-s − 10.7i·37-s + 4.43·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.67i·7-s − 0.333·9-s + 1.25·11-s − 1.23i·13-s − 0.327i·17-s − 0.620·19-s − 0.968·21-s + 0.281i·23-s − 0.192i·27-s + 1.34·29-s − 0.179·31-s + 0.726i·33-s − 1.77i·37-s + 0.710·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.731432115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.731432115\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 4.43iT - 7T^{2} \) |
| 11 | \( 1 - 4.17T + 11T^{2} \) |
| 13 | \( 1 + 4.43iT - 13T^{2} \) |
| 17 | \( 1 + 1.35iT - 17T^{2} \) |
| 19 | \( 1 + 2.70T + 19T^{2} \) |
| 23 | \( 1 - 1.35iT - 23T^{2} \) |
| 29 | \( 1 - 7.25T + 29T^{2} \) |
| 37 | \( 1 + 10.7iT - 37T^{2} \) |
| 41 | \( 1 - 2.17T + 41T^{2} \) |
| 43 | \( 1 + 9.46iT - 43T^{2} \) |
| 47 | \( 1 + 4.11iT - 47T^{2} \) |
| 53 | \( 1 + 12.8iT - 53T^{2} \) |
| 59 | \( 1 + 5.25T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 1.14iT - 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 10.4iT - 73T^{2} \) |
| 79 | \( 1 - 2.05T + 79T^{2} \) |
| 83 | \( 1 - 6.99iT - 83T^{2} \) |
| 89 | \( 1 - 2.55T + 89T^{2} \) |
| 97 | \( 1 + 15.9iT - 97T^{2} \) |
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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
−7.79801210048235873961324769372, −6.84733172247686420136853732825, −6.14949496032493713167440145299, −5.55044105911736896413757627190, −5.06509288794127035285670999314, −4.10919318636213870434773771827, −3.37253900314929970499770396643, −2.61260227592534288975844731926, −1.83935837526820076786733427029, −0.42087764935850210725860364030,
1.10936554443113685755217049704, 1.40868284061869175461861496665, 2.65938344221767519781320877711, 3.62713802287587036280635301689, 4.40508441655140235341308699452, 4.58957781153769338996630889155, 6.10364683705643004968416054796, 6.56035721501968794121808872907, 6.91147834140534909165663228235, 7.68330816185553293170799286123
